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King Fahd University of Petroleum and Minerals
Information and Computer Science Department
ICS 583: Pattern Recognition
Homework Assignment #1
(Due Sunday March 7, 2010 at 6:30pm)
Answer the following questions from the book: Alberto Leon-Garcia, “Probability,
Statistics, and Random Processes for Electrical Engineering”, Third Edition, 2009.
1. (20 points) Question 2.5 from pages 81-82:
A desk drawer contains six pens, four of which are dry.
a. (5 points) The pens are selected at random one by one until a good pen is
found. The sequence of test results is noted. What is the sample space?
b. (3 points) Suppose that only the number, and not the sequence, of pens tested
in part a is noted. Specify the sample space.
c. (7 points) Suppose that the pens are selected one by one and tested until both
good pens have been identified, and the sequence of test results is noted. What
is the sample space?
d. (5 points) Specify the sample space in part c if only the number of pens tested
is noted.
2. (10 points) Question 2.8 from page 82:
A number U is selected at random from the unit interval. Let the events A and B be:
A = “U differs from 1/2 by more than 1/4” and B = “1 – U is less than 1/2.” Find the
events A  B, Ac  B, A  B c .
3. (30 points) Question 2.34 from page 85 and 2.69 from page 88:
A number x is selected at random in the interval [-1, 2]. Let the events A  x  0,
B  x  0.5  0.5 and C  x  0.75
a. (9 points) Find the probabilities of A, B, A  B, and A C.
b. (9 points) Find the probabilities of A  B, A  C , and A  B  C.
c. (12 points) Find the probabilities: P[ A | B], P[ B | C ], P[ A | C C ], P[ B | C C ] .
4. (15 points) Question 2.71 from page 88.
Find the probability that two or more students in a class of 20 students have the same
birthday. Hint: Use P[ AC ]  1  P[ A]. How big should the class be so that the
probability that two or more students have the same birthday is 1/2?
5. (25 points) Question 3.13 from page 132 and 3.27 from page 133.
Let X be a random variable with pmf (i.e. pdf) pk  c / k 2 for k  1,2,...
a. (5 points) Estimate the value of c numerically. Note that the series converges.
b. (5 points) Find P[ X  4]
c. (5 points) Find P[6  X  8]
d. (10 points) Find E[X] and VAR[X].